Heads-up NL hold'em with extremely small stacks

I have not made a lot of money playing poker. I've won a little playing NL hold'em tournaments, though. I think I do slightly better in the early stages of the tournaments, because I'm a little too passive to play well in short-handed situations. Sometimes when I've made it to the final table, I see people that move all-in preflop quite often. I've even seen a player that moved all-in on every hand after he became chip leader. This might very well be a horrible play in a ten-handed game, but can be somewhat more effective when the blinds get big and the game becomes shorthanded.

How do you counter this play? I decided to concentrate on heads-up play in this article. I've made some simplifications. I assume that you're up against an opponent that moves all-in every time he's on the button (first to act pre-flop). How should you play when you are on the button yourself? To make the calculations easier, I assume that you limp in every time you're on the button. This is probably not the best play under normal circumstances, but let's ignore that. Because you play the same way every time, your opponent will not know if you have a good or a bad hand. I assume that he moves all-in every time you limp in.

Based on this information about your opponent, you have to decide whether or not to call when he makes a push. Now I'm going to study games where the total amount of chips are less than 10 big blinds. I'm going to assume that both players have an amount of chips equal to a certain (integer) number of big blinds.

I'm using a table that can be found at this internet address:

http://gocee.com/poker/HE_Value.htm

Because your opponent moves all-in on every hand, you assume that he has a random hand when he moves all-in. The table shows the win% of different hands against random hands. The worst hand you can have against a random hand (at least when all the money goes in pre-flop) is 32 offsuit. According to the table, it wins 31,2 percent of the time.

If the total amount of chips equals 3 big blinds, one of the players will be all-in preflop anyway, so that's not a very interesting scenario. When the total amount of chips equals 4 big blinds and both players has the same amount of chips, you can ask yourself if you're pot commited to call a pre-flop push once you have put one big blind in the middle. Even if you have 32 offsuit, it wins 31,2 percent of the time against a random hand. By folding, you have a 50 % chance to double up, and 25 % chance to quadruple your amount of chips and win. This is considerably less than the win% of 32 offsuit against a random hand, so you should call with 32 offsuit when this opponent pushes pre-flop in this scenario.

If the total amount of chips equals 5 big blinds, there are two interesting scenarios. In one scenario, you can afford to pay two big blinds. Your opponent has 1,5 times as many chips as you. If you fold, you have 1/5 of the remaining chips. This means that you have a 20 % chance of winning if you push on every hand. If you have 32 offsuit, and calls the push you'll have a 31,2 % chance of getting 4/5 of the chips on the table. Once you've gotten all those chips, you have a 80 % chance of winning all the chips. Your total chance of winning is (31,2 % * 0,8) = 24,96 %. You can see that a push is more profitable than a fold.

In the other scenario, you can afford 3 big blinds. Similar calculations as above show that a call with 32 offsuit gives you a 44,96 % chance of winning, while a fold gives you a 40 % chance. You should still call anytime your opponent pushes.

If the total amount of chips equals 6 big blinds, things are starting to get more interesting. Let's say that you have half as many chips as your opponent. If you fold, you'll be all-in on the next hand with 1/6 (16,67 % chance of winning) of the total amount of chips on the table. If you call with 32 offsuit, you'll win 4/6 of the chips 31,2 % of the time. Your total chance of winning is (31,2 % * 4/6) = 20,8 %. A push is most profitable with those stack sizes. Similar calculations show that a push with 32 offsuit when you have twice as many chips as your opponent give you a 54,13 % chance of winning, while a fold only gives you a 50 % chance.

Assume that you have enough chips to pay 3 big blinds (the same as your opponent). If you call with 32 offsuit, you have a 31,2 % chance of winning. If you fold, you still have 2/6 of the chips on the table. This gives you a 33,3 percent chance of winning. This means that you should fold 32 offsuit and 42 offsuit (32,5 % chance of winning) after your opponent moves all-in preflop.

I wrote earlier that the chance of winning when you fold 32 offsuit with twice as many chips as your opponent is 50 %. This was a small idealization. Folding in that situation results in your opponent getting the same amount of chips as yourself. When you have the same amount of chips as your opponent, you have an edge over him. This is because he commits all his chips on any hand, while you fold 32 offsuit and 42 offsuit. This added edge, means that you actually have about a 50,03 percent chance of winning if you fold. It's still smaller than the 54,13 % chance of winning that you get if push, though.

So far, the calculations have not been too difficult. It gets harder when the total amount of chips equals 7 big blinds. I've come to the conclusion that you should call when you have 2/7 or 5/7 of the total amount of chips, and fold when you have 3/7 or 4/7 of the total amount of chips and get dealt 32 offsuit or 42 offsuit. I'm not going to publish any numbers here now. It gets insanely difficult to calculate the exact numbers with the method I'm using, because you can get all kinds of strange developments. Every time you have 3/7 or 4/7 of the total amount of chips, you have to calculate your edge. My method is not well suited to do this. To illustrate a possible development, you can imagine that the number of big blinds you can afford evolves like this:

3(p)-6(p)-5(p)-3(f)-2(p)-4(f)-3(f)-2(p)-4(p)-1(p)-2(p)-4(f)-3(p)-0(you lose)

p means that you call the push, f that you fold (you start with 3 big blinds, call and win. Then you have 6 big blinds, but lose the next pot. You have 5 big blinds left, and lose one more time. Now you have 3 big blinds left, and get 32 offsuit. You fold, have 2 big blinds left and so on).
You can see from all the folding that you've been dealt 32o or 42o extraordinarily often in this example. No wonder that you lost.

I'm going to look more closely at this, and might write more later. It might be more interesting to look at stack sizes where it becomes profitable to fold pocket pairs. At some point, it would be correct to fold pocket kings against this opponent. You'd probably have to afford many thousand big blinds to make it profitable to fold kings, though.

Poker theory, by THEjDonk

I'm known as THEjDonk on a couple of poker sites. I've written a couple of poker articles, and I'm going to present them here. Some or most of them might not be directly applicable to any poker game you're likely to encounter. They should be somewhat interesting for people that enjoy reading theoretical material, though.